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Integral models of representations of current groups. (English) Zbl 1162.22019
Funct. Anal. Appl. 42, No. 1, 19-27 (2008); translation from Funkts. Anal. Prilozh. 42, No. 1, 22-32 (2008).
Authors’ summary: We suggest a new construction of nonlocal representations of the current group. Instead of the Fock space, which is usually used in this situation, we consider the direct integral of countable tensor products of representations over the trajectories of some stochastic process. The construction substantially uses the invariance of the so-called infinite-dimensional Lebesgue measure.

##### MSC:
 2.2e+66 Infinite-dimensional Lie groups and their Lie algebras: general properties 2.2e+68 Loop groups and related constructions, group-theoretic treatment
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##### References:
 [1] A. M. Vershik and M. I. Graev, ”Structure of the complementary series and special representations of the groups O(n, 1) and U(n, 1),” Uspekhi Mat. Nauk, 61:5 (2006), 3–88; English transl.: Russian Math. Surveys, 61:5 (2006), 799–884. · Zbl 1148.22017 [2] M. I. Graev and A. M. Vershik, ”The basic representation of the current group O(n, 1)X in the L 2 space over the generalized Lebesgue measure,” Indag. Math., 16:3–4 (2005), 499–529. · Zbl 1147.22013 [3] I. M. Gelfand, M. I. Graev, and A. M. Vershik, ”Models of representations of current groups,” in: Representations of Lie groups and Lie algebras, Akad. Kiadó, Budapest, 1985, 121–179. [4] A. M. Vershik, I. M. Gelfand, and M. I. Graev, ”Irreducible representations of the group G X and cohomology,” Funkts. Anal. Prilozhen., 8:2 (1974), 67–69; English transl.: Functional Anal. Appl., 8:2 (1974), 151–153. [5] A. M. Vershik, I. M. Gelfand, and M. I. Graev, ”Representations of the group SL(2, R), where R is a ring of functions,” Uspekhi Mat. Nauk, 28:5 (1973), 83–128; English transl.: Russian Math. Surveys, 28:5 (1973), 87–132. [6] A. M. Vershik, I. M. Gelfand, and M. I. Graev, ”The commutative model of representation of the group of flows SL(2, $$\mathbb{R}$$)X connected with a unipotent subgroup,” Funkts. Anal. Prilozhen., 17:2 (1983), 70–72; English transl.: Functional Anal. Appl., 17:2 (1983), 137–139. · Zbl 0522.46017 [7] A. M. Vershik, ”Does there exist the Lebesgue measure in the infinite-dimensional space?,” Trudy Mat. Inst. Steklov, 259 (2007), 256–281. · Zbl 1165.28003 [8] A. M. Vershik and S. I. Karpushev, ”Cohomology of groups in unitary representations, the neighborhood of the identity, and conditionally positive definite functions,” Mat. Sb., 119:4 (1982), 521–533; English transl.: Math. USSR Sb., 47 (1984), 513–526. · Zbl 0513.43009 [9] A. M. Vershik and N. V. Tsilevich, ”Fock factorizations and decompositions of the L 2 spaces over general Levy processes,” Uspekhi Mat. Nauk, 58:3 (351) (2003), 3–50; English transl.: Russian Math. Surveys, 58:3 (2003), 427–472. · Zbl 1060.46056 [10] N. Tsilevich, A. Vershik, and M. Yor, ”An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process,” J. Funct. Anal., 185:1 (2001), 274–296. · Zbl 0990.60053 [11] H. Araki, ”Factorizable representations of current algebra,” Publ. RIMS Kyoto Univ. Ser. A, 5:3 (1970), 361–422. · Zbl 0238.22014 [12] A. Perelomov, Generalized Coherent States and Their Applications, Berlin, Springer-Verlag, 1986. · Zbl 0605.22013 [13] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. II (based, in part, on notes left by H. Bateman), McGraw-Hill Book Company, New York-Toronto-London, 1953. · Zbl 0051.30303
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